Method and System for Utilizing Givens Rotation Expressions for Asymmetric Beamforming Matrices in Explicit Feedback Information

ABSTRACT

Aspects of a method and system for utilizing Givens rotation expressions for asymmetric beamforming matrices in explicit feedback information are presented. In one aspect of the invention, Givens matrices may be utilized to reduce a quantity of information communicated in explicit feedback information via an uplink RF channel. The explicit feedback information may include specifications for a feedback beamforming matrix that may be utilized when transmitting signals via a corresponding downlink RF channel. The feedback beamforming matrix may represent a rotated version of an un-rotated matrix. The Givens matrices may be utilized to apply one or more Givens rotations to un-rotated matrix. The feedback beamforming matrix may be computed based on a matrix product of a plurality of Givens matrices. The feedback beamforming matrix may be encoded utilizing fewer bits than may be required to encode the un-rotated matrix.

CROSS REFERENCE TO PRIORITY APPLICATIONS

The present U.S. Utility Patent Application claims priority pursuant to35 U.S.C. §120, as a continuation, to the following U.S. Utility PatentApplication which is hereby incorporated herein by reference in itsentirety and made part of the present U.S. Utility Patent Applicationfor all purposes:

-   -   1. U.S. Utility application Ser. No. 11,327,690, entitled,        “Method and System for Utilizing Givens Rotation Expressions for        Asymmetric Beamforming Matrices in Explicit Feedback        Information,” filed Jan. 6, 2006, pending, which claims priority        pursuant to 35 U.S.C. §119(e) the following U.S. Provisional        Patent Application which is hereby incorporated herein by        reference in its entirety and made part of the present U.S.        Utility Patent Application for all purposes:        -   a. U.S. Provisional Application Ser. No. 60/734,514,            entitled, “Method and System for Givens Rotation Expression            for Asymmetric Beamforming Matrix in the Explicit Feedback,”            filed Nov. 7, 2005, now expired.

CROSS REFERENCE TO RELATED APPLICATIONS

This application makes reference to:

U.S. application Ser. No. 11/327,752 filed Jan. 6, 2006; and U.S.application Ser. No. 11/052,389 filed Feb. 7, 2005.

FIELD OF THE INVENTION

Certain embodiments of the invention relate to wireless communications.More specifically, certain embodiments of the invention relate to amethod and system for utilizing Givens rotation expressions forasymmetric beamforming matrices in explicit feedback information.

BACKGROUND OF THE INVENTION

Multiple input multiple output (MIMO) systems are wirelesscommunications systems are specified in resolution 802.11 from theInstitute of Electrical and Electronics Engineers (IEEE). A MIMO systemthat receives a signal Y may compute a channel estimate matrix, H, basedon the received signal. The signal may comprise information generatedfrom a plurality of information sources. Each such information sourcemay be referred to as a spatial stream. A transmitting MIMO system mayutilize a plurality of transmitting antennas when transmitting thesignal Y. A receiving MIMO system may utilize a plurality of receivingantennas when receiving the signal Y. The channel estimate matrix for adownlink RF channel, H_(down), may describe a characteristic of thewireless transmission medium in the transmission path from atransmitter, to a receiver. The channel estimate for an uplink RFchannel, H_(up), may describe a characteristic of the wirelesstransmission medium in the transmission path from the receiver to thetransmitter.

According to the principle of reciprocity, a characteristic of thewireless transmission medium in the transmission path from thetransmitter to the receiver may be assumed to be identical to acorresponding characteristic of the wireless transmission medium in thetransmission path from the receiver to the transmitter. However, thechannel estimate matrix H_(down) may not be equal to a correspondingchannel estimate matrix for an uplink RF channel H_(up). For example, anoise level, for example an ambient noise level, in the vicinity of thetransmitter may differ from a noise level in the vicinity of thereceiver. Similarly, an interference level, for example electro-magneticinterference due to other electro-magnetic devices, in the vicinity ofthe transmitter may differ from an interference level in the vicinity ofthe receiver. At a transmitter, or receiver, there may also beelectrical cross-coupling, for example leakage currents, betweencircuitry associated with a receiving antenna, or a transmittingantenna, and circuitry associated with another receiving antenna, oranother transmitting antenna.

The principle of reciprocity, wherein it may be assumed thatH_(up)=H_(down), may also be based on the assumption that specificantennas at a transmitter or receiver are assigned for use astransmitting antennas, and/or assigned for use as receiving antennas. Atthe transmitter, a number of receiving antennas, NRX, utilized at thereceiver may be assumed. At the receiver, a number of transmittingantennas, NTX, utilized at the transmitter may be assumed. If theassignments of at least a portion of the antennas at the transmitter arechanged, the corresponding channel estimate matrix H′_(up) may not beequal H_(down). Similarly, if the assignments of at least a portion ofthe antennas at the receiver are changed, the corresponding channelestimate matrix H′_(down) may not be equal H_(up). Consequently, afterreassignment of antennas at the transmitter and/or receiver, theprinciple of reciprocity may not be utilized to characterizecommunications between the transmitter and the receiver when H_(up) doesnot equal H′_(down), when H′_(up) does not equal H_(down), or whenH′_(up) does not equal H′_(down).

The principle of reciprocity may enable a receiving wireless local areanetwork (WLAN) device A to receive a signal Y from a transmitting WLANdevice B, and to estimate a channel estimate matrix H_(down) for thetransmission path from the transmitting WLAN device B to the receivingWLAN device A. Based on the channel estimate matrix H_(down), the WLANdevice A may transmit a subsequent signal Y, via an uplink RF channel,to the WLAN device B based on the assumption that the channel estimatematrix H_(up) for the transmission path from the transmitting WLANdevice A to the receiving WLAN device B may be characterized by therelationship H_(up)=H_(down). When the WLAN devices A and B are MIMOsystems, corresponding beamforming matrices may be configured andutilized for transmitting and/or receiving signals at each WLAN device.

Beamforming is a method for signal processing that may allow atransmitting MIMO system to combine a plurality of spatial streams in atransmitted signal Y. Beamforming is also a method for signal processingthat may allow a receiving MMO system to separate individual spatialstreams in a received signal Y.

As a result of a failure of an assumed condition for the principle ofreciprocity, a beamforming matrix at the transmitting WLAN device,and/or a beamforming matrix at the receiving WLAN device, may beconfigured incorrectly. In a transmitted signal Y, from the perspectiveof a signal associated with an i^(th) spatial stream, a signalassociated with a j^(th) spatial stream may represent interference ornoise. Incorrect configuration of one or more beamforming matrices mayreduce the ability of the receiving WLAN device to cancel interferencebetween an i^(th) spatial stream and a j^(th) spatial stream.Consequently, the received signal Y may be characterized by reducedsignal to noise ratios (SNR). There may also be an elevated packet errorrate (PER) when the receiving WLAN device decodes information containedin the received signal Y. This may, in turn, result in a reducedinformation transfer rate, as measured in bits/second, forcommunications between the transmitting WLAN device and the receivingWLAN device.

Further limitations and disadvantages of conventional and traditionalapproaches will become apparent to one of skill in the art, throughcomparison of such systems with some aspects of the present invention asset forth in the remainder of the present application with reference tothe drawings.

BRIEF SUMMARY OF THE INVENTION

A system and/or method for utilizing Givens rotation expressions forasymmetric beamforming matrices in explicit feedback information,substantially as shown in and/or described in connection with at leastone of the figures, as set forth more completely in the claims.

These and other advantages, aspects and novel features of the presentinvention, as well as details of an illustrated embodiment thereof, willbe more fully understood from the following description and drawings.

BRIEF DESCRIPTION OF SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 is a block diagram of an exemplary system for wireless datacommunications, which may be utilized in connection with an embodimentof the invention.

FIG. 2 is a block diagram of an exemplary MIMO system that may beutilized in connection with an embodiment of the invention.

FIG. 3 is an exemplary diagram illustrating beamforming that may beutilized in connection with an embodiment of the invention.

FIG. 4 is an exemplary functional block diagram of transceivercomprising a transmitter and a receiver in a MIMO system, which may beutilized in accordance with an embodiment of the invention.

FIG. 5 is a flowchart illustrating exemplary steps for utilizing Givensrotations for beamforming matrices in explicit feedback information, inaccordance with an embodiment of the invention

DETAILED DESCRIPTION OF THE INVENTION

Certain embodiments of the invention relate to a method and system forutilizing Givens rotation expressions for asymmetric beamformingmatrices in explicit feedback information. In one aspect of theinvention, Givens matrices may be utilized to reduce a quantity ofinformation communicated in explicit feedback information via an uplinkRF channel. The explicit feedback information may comprisespecifications for a feedback beamforming matrix that may be utilizedwhen transmitting signals via a corresponding downlink RF channel. Thefeedback beamforming matrix may represent a rotated version of anun-rotated beamforming matrix. The Givens matrices may be utilized toapply one or more Givens rotations to un-rotated beamforming matrix. Thefeedback beamforming matrix may be computed based on a matrix product ofa plurality of Givens matrices. The feedback beamforming matrix maycomprise comparable information content to the un-rotated beamformingmatrix. The feedback beamforming matrix may be encoded utilizing fewerbits than may be required to encode the un-rotated beamforming matrix.Various embodiments of the invention may be utilized in wirelesscommunications systems, including wireless data communications systems.

Explicit feedback may be utilized to reduce the likelihood that signalsY will be transmitted incorrectly during beamforming. Explicit feedbackcomprises a mechanism by which a receiving WLAN device A may compute achannel estimate matrix H_(down). The computed channel estimate matrixmay be utilized to compute feedback information that may be utilized bythe WLAN device B to configure a beamforming matrix associated with thedownlink RF channel. The feedback information may be communicated fromthe WLAN device A to the WLAN device B via the uplink RF channel. Thefeedback information may comprise a full description of the downlink RFchannel. A full description of the downlink RF channel may comprise afrequency channel estimate for each subcarrier frequency associated witheach spatial stream contained in the downlink RF channel. Each frequencychannel estimate may be characterized in a Cartesian coordinate format,or in a polar coordinate format. In the Cartesian coordinate format,each frequency channel estimate may be represented by an in-phase (I)amplitude component, and a quadrature (Q) phase amplitude component. Inthe polar coordinate format, each frequency channel estimate may berepresented by a magnitude (p) component, and an angle (θ) component.

In some conventional systems, a large quantity of feedback informationmay be communicated in the feedback direction. RF channel bandwidth thatis dedicated for use in communicating feedback information may reducethe quantity of other types of information that may be communicatedbetween the WLAN device A and the WLAN device B within a given timeinterval. The feedback information may be referred to as overhead withinthe uplink RF channel. The overhead may reduce the information transferrate between the WLAN device A and the WLAN device B associated withnon-feedback information.

FIG. 1 is a block diagram of an exemplary system for wireless datacommunications, which may be utilized in connection with an embodimentof the invention. With reference to FIG. 1 there is shown a distributionsystem (DS) 110, an extended service set (ESS) 120, and an IEEE 802.xLAN 122. The ESS 120 may comprise a first basic service set (BSS) 102,and a second BSS 112. The first BSS 102 may comprise a first 802.11 WLANstation 104, a second 802.11 WLAN station 106, and an access point (AP)108. The second BSS 112 may comprise a first 802.11 WLAN station 114, asecond 802.11 WLAN station 116, and an access point (AP) 118. The IEEE802 LAN 122 may comprise a LAN station 124, and a portal 126. An IEEE802.11 WLAN station, or IEEE 802.11 WLAN device, is a WLAN system thatmay be compliant with at least a portion of the IEEE 802.11 standard.

A WLAN is a communications networking environment that comprises aplurality of WLAN devices that may communicate wirelessly via one ormore uplink and/or downlink RF channels. The BSS 102 or 112 may be partof an IEEE 802.11 WLAN that comprises at least 2 IEEE 802.11 WLANstations, for example, the first 802.11 WLAN station 104, the second802.11 WLAN station 106, and the AP 108, which may be members of the BSS102. Non-AP stations within BSS 102, the first 802.11 WLAN station 104,and the second 802.11 WLAN station 106, may individually form anassociation with the AP 108. An AP, such as AP 108, may be implementedas an Ethernet switch, bridge, or other device in a WLAN, for example.Similarly, non-AP stations within BSS 112, the first 802.11 WLAN station114, and the second 802.11 WLAN station 116, may individually form anassociation with the AP 118. Once an association has been formed betweena first 802.11 WLAN station 104 and an AP 108, the AP 108 maycommunicate reachability information about the first 802.11 WLAN station104 to other APs associated with the ESS 120, such as AP 118, andportals such as the portal 126. The WLAN station 104 may subsequentlycommunicate information wirelessly via the BSS 102. In turn, the AP 118may communicate reachability information about the first 802.11 WLANstation 104 to stations in BSS 112. The portal 126, which may beimplemented as, for example, an Ethernet switch or other device in aLAN, may communicate reachability information about the first 802.11WLAN station 104 to stations in LAN 122 such as the 802 LAN station 124.The communication of reachability information about the first 802.11WLAN station 104 may enable WLAN stations that are not in BSS 102, butare associated with ESS 120, to communicate wirelessly with the first802.11 WLAN station 104.

The DS 110 may provide an infrastructure which enables a first 802.11WLAN station 104 in one BSS 102, to communicate wirelessly with a first802.11 WLAN station 114 in another BSS 112. The DS 110 may also enable afirst 802.11 WLAN station 104 in one BSS 102 to communicate with an 802LAN station 124 in an IEEE 802 LAN 122, implemented as, for example awired LAN. The AP 108, AP 118, or portal 126 may provide a means bywhich a station in a BSS 102, BSS 112, or LAN 122 may communicateinformation via the DS 110. The first 802.11 WLAN station 104 in BSS 102may communicate information wirelessly to a first 802.11 WLAN station114 in BSS 112 by transmitting the information wirelessly to AP 108,which may transmit the information via the DS 110 to AP 118, which inturn may transmit the information wirelessly to station 114 in BSS 112.The first 802.11 WLAN station 104 may communicate information wirelesslyto the 802 LAN station 124 in LAN 122 by transmitting the informationwirelessly to AP 108, which may transmit the information via the DS 110to the portal 126, which in turn may transmit the information to the 802LAN station 124 in LAN 122. The DS 110 may utilize wirelesscommunications via an RF channel, wired communications, such as IEEE 802Ethernet, or a combination thereof.

A WLAN station or AP may utilize one or more transmitting antennas, andone or more receiving antennas when communicating information. A WLANstation or AP that utilizes a plurality of transmitting antennas and/ora plurality of receiving antennas may be referred to as a multiple inputmultiple output (MIMO) system.

FIG. 2 is a block diagram of an exemplary MIMO system that may beutilized in connection with an embodiment of the invention. Withreference to FIG. 2 there is shown a baseband processor 272, atransceiver 274, an RF front end 280, a plurality of receiving antennas276 a, . . . , 276 n, and a plurality of transmitting antennas 278 a, .. . , 278 n. The transceiver 274 may comprise a processor 282, areceiver 284, and a transmitter 286.

The processor 282 may perform digital receiver and/or transmitterfunctions in accordance with applicable communications standards. Thesefunctions may comprise, but are not limited to, tasks performed at lowerlayers in a relevant protocol reference model. These tasks may furthercomprise the physical layer convergence procedure (PLCP), physicalmedium dependent (PMD) functions, and associated layer managementfunctions. The baseband processor 272 may perform functions inaccordance with applicable communications standards. These functions maycomprise, but are not limited to, tasks related to analysis of datareceived via the receiver 284, and tasks related to generating data tobe transmitted via the transmitter 286. These tasks may further comprisemedium access control (MAC) layer functions as specified by pertinentstandards.

The receiver 284 may perform digital receiver functions that maycomprise, but are not limited to, fast Fourier transform processing,beamforming processing, equalization, demapping, demodulation control,deinterleaving, depuncture, and decoding. The transmitter 286 mayperform digital transmitter functions that comprise, but are not limitedto, coding, puncture, interleaving, mapping, modulation control, inversefast Fourier transform processing, beamforming processing. The RF frontend 280 may receive analog RF signals via antennas 276 a, . . . , 276 n,converting the RF signal to baseband and generating a digital equivalentof the received analog baseband signal. The digital representation maybe a complex quantity comprising I and Q components. The RF front end280 may also transmit analog RF signals via an antenna 278 a, . . . ,278 n, converting a digital baseband signal to an analog RF signal.

In operation, the processor 282 may receive data from the receiver 284.The processor 282 may communicate received data to the basebandprocessor 272 for analysis and further processing. The basebandprocessor 272 may generate data to be transmitted via an RF channel bythe transmitter 286. The baseband processor 272 may communicate the datato the processor 282. The processor 282 may generate a plurality of bitsthat are communicated to the receiver 284. The processor 282 maygenerate signals to control the operation of the modulation process inthe transmitter 286, and of the demodulation process in the receiver284.

FIG. 3 is an exemplary diagram illustrating beamforming that may beutilized in connection with an embodiment of the invention. Referring toFIG. 3 there is shown a transmitting mobile terminal 302, a receivingmobile terminal 322, and a plurality of RF channels 342. Thetransmitting mobile terminal 302 comprises a transmit filter coefficientblock V 304, a first source signal s₁ 306, a second source signal s₂308, a third source signal s₃ 310, and a plurality of transmittingantenna 312, 314, and 316.

In operation, the transmitting antenna 312 may enable transmission of asignal x₁, the transmitting antenna 314 may transmit a signal x₂, andthe transmitting antenna 316 may transmit a signal x₃. In beamformingeach transmitted signal x₁, x₂, and x₃ may be a function of a weightedsummation of at least one of the plurality of the source signals s₁, s₂,and s₃. The weights may be determined by the transmit filter coefficientblock V such that: X=VS, where S may be represented by a 3×1 matrix {s₁,s₂, s₃}, V may be represented by a 3×3 matrix {{v₁₁, v₁₂, v₁₃}{v₂₁, v₂₂,v₂₃}{v₃₁, v₃₂, v₃₃}}, and X may be represented by a 3×1 matrix {x₁, x₂,x₃}. The receiving mobile terminal 322 comprises a receive filtercoefficient block U* 324, a first destination signal y ₁ 326, a seconddestination signal y ₂ 328, a third destination signal y ₃ 330, and aplurality of receiving antenna 332, 334, and 336. The receiving antenna332 may receive a signal y₁, the receiving antenna 334 may receive asignal y₂, and the receiving antenna 336 may receive a signal y₃. Thecharacteristics of the plurality of RF channels 342 utilized forcommunication between the transmitting mobile terminal 302, and thereceiving mobile terminal 322 may be represented mathematically by atransfer coefficient matrix H.

The plurality of received signals y₁, y₂, y₃, may be expressed as afunction of the plurality of transmitted signals x₁, x₂, x₃, and thetransfer coefficient matrix H in the following equation:

Y=HX+N, where  equation [1]

Y={y₁, y₂, y₃} may be represented as a 3×1 matrix; X may be representedas a 3×1 matrix {x_(i), x₂, x₃}; H may be represented as a 3×3 matrix{{h₁₁, h₁₂, h₁₃}{h₂₁, h₂₂, h₂₃}{h₃₁, h₃₂, h₃₃}}; and N may berepresented as is a 3×1 matrix {n₁, n₂, n₃}. The matrix N may representrandom noise that may exist in the communication medium between thetransmitting station 302 and the receiving station 322.

In a MIMO system that utilizes beamforming, the values associated withthe matrices V, H, and U* may be such that the value of the transmittedsignal s_(i) may be approximately equal to the value of thecorresponding received signal y _(i), where the index i may indicate ani^(th) spatial stream. The receiver 322 may compute a channel estimate,H_(down), for a downlink RF channel based on a received signal from thetransmitter 302. The transmitter 302 may compute a channel estimate,H_(up), for an uplink RF channel based on a received signal from thereceiver 322. In the absence of explicit feedback information receivedfrom the receiver 322, the transmitter may assume channel reciprocity.The coefficient block V 304 may set values associated with the matrix V,utilized for communications via the downlink RF channel, based on thechannel estimate H_(up).

When the receiver 322 sends explicit feedback information to thetransmitter 302 via the uplink RF channel, the receiver 322 may computevalues associated with a matrix V that may be based on the channelestimate H_(down). The receiver 322 may subsequently communicate thecomputed matrix V to the transmitter 302 via the uplink RF channel. Thetransmitter 302 may subsequently set values associated with the matrixV, utilized for communications via the downlink RF channel, based on thechannel estimate H_(down).

Various embodiments of the invention may reduce the quantity of explicitfeedback information that may be communicated from the receiver 322 tothe transmitter 302 via the uplink RF channel. In one aspect of theinvention, the computed matrix V may be represented based on a productof Givens matrices:

V=G ₁(ψ₁)G ₂(ψ₂)D ₁(φ₁,φ₂, . . . , φ_(N))  equation [2]

where G_(j)(ψ_(j)) may represent a Givens matrix associated with arotation angle ψ_(j), and D₁(φ_(k)) may represent a diagonal matrix withphase shift angles φ_(k).

The explicit feedback information communicated by the receiver 322 maycomprise encoded rotation angles ψ_(j). The transmitter 302 may utilizethe encoded rotation angles to reconstruct the matrix V based on theexplicit feedback information.

FIG. 4 is an exemplary functional block diagram of transceivercomprising a transmitter and a receiver in a MIMO system, which may beutilized in accordance with an embodiment of the invention. FIG. 4 showstransceiver comprising a transmitter 400, a receiver 401, a processor440, a baseband processor 442, a plurality of transmitter antennas 415 a. . . 415 n, and a plurality of receiver antennas 417 a . . . 417 n. Thetransmitter 400 may comprise a coding block 402, a puncture block 404,an interleaver block 406, a plurality of mapper blocks 408 a . . . 408n, a plurality of inverse fast Fourier transform (IFFT) blocks 410 a . .. 410 n, a beamforming V matrix block 412, and a plurality of digital toanalog (D to A) conversion and antenna front end blocks 414 a . . . 414n. The receiver 401 may comprise a plurality of antenna front end andanalog to digital (A to D) conversion blocks 416 a . . . 416 n, abeamforming U* matrix block 418, a plurality of fast Fourier transform(FFT) blocks 420 a . . . 420 n, a channel estimates block 422, anequalizer block 424, a plurality of demapper blocks 426 a . . . 426 n, adeinterleaver block 428, a depuncture block 430, and a Viterbi decoderblock 432.

The variables V and U* in beamforming blocks 412 and 418 respectivelyrefer to matrices utilized in the beamforming technique. U.S.application Ser. No. 11/052,389 filed Feb. 7, 2005, provides a detaileddescription of Eigenbeamforming, which is hereby incorporated herein byreference in its entirety.

The processor 440 may perform digital receiver and/or transmitterfunctions in accordance with applicable communications standards. Thesefunctions may comprise, but are not limited to, tasks performed at lowerlayers in a relevant protocol reference model. These tasks may furthercomprise the physical layer convergence procedure (PLCP), physicalmedium dependent (PMD) functions, and associated layer managementfunctions. The baseband processor 442 may similarly perform functions inaccordance with applicable communications standards. These functions maycomprise, but are not limited to, tasks related to analysis of datareceived via the receiver 401, and tasks related to generating data tobe transmitted via the transmitter 400. These tasks may further comprisemedium access control (MAC) layer functions as specified by pertinentstandards.

In the transmitter 400, the coding block 402 may transform receivedbinary input data blocks by applying a forward error correction (FEC)technique, for example, binary convolutional coding (BCC). Theapplication of FEC techniques, also known as “channel coding”, mayimprove the ability to successfully recover transmitted data at areceiver by appending redundant information to the input data prior totransmission via an RF channel. The ratio of the number of bits in thebinary input data block to the number of bits in the transformed datablock may be known as the “coding rate”. The coding rate may bespecified using the notation i_(b)/t_(b), where t_(b) represents thetotal number of bits that comprise a coding group of bits, while i_(b)represents the number of information bits that are contained in thegroup of bits t_(b). Any number of bits t_(b)−i_(b) may representredundant bits that may enable the receiver 401 to detect and correcterrors introduced during transmission. Increasing the number ofredundant bits may enable greater capabilities at the receiver to detectand correct errors in information bits. The penalty for this additionalerror detection and correction capability may result in a reduction inthe information transfer rates between the transmitter 400 and thereceiver 401. The invention is not limited to BCC, and any one of aplurality of coding techniques, for example, Turbo coding or low densityparity check (LDPC) coding, may also be utilized.

The puncture block 404 may receive transformed binary input data blocksfrom the coding block 402 and alter the coding rate by removingredundant bits from the received transformed binary input data blocks.For example, if the coding block 402 implemented a ½ coding rate, 4 bitsof data received from the coding block 402 may comprise 2 informationbits, and 2 redundant bits. By eliminating 1 of the redundant bits inthe group of 4 bits, the puncture block 404 may adapt the coding ratefrom ½ to ⅔. The interleaver block 406 may rearrange bits received in acoding rate-adapted data block from the puncture block 404 prior totransmission via an RF channel to reduce the probability ofuncorrectable corruption of data due to burst of errors, impactingcontiguous bits, during transmission via an RF channel. The output fromthe interleaver block 406 may also be divided into a plurality ofstreams where each stream may comprise a non-overlapping portion of thebits from the received coding rate-adapted data block. Therefore, for agiven number of bits in the coding rate-adapted data block, b_(db), agiven number of streams from the interleaver block 406, n_(st), and agiven number of bits assigned to an individual stream i by theinterleaver block 406, b_(st)(i):

$\begin{matrix}{b_{db} = {\sum\limits_{i = 0}^{n_{st} - 1}{b_{st}(i)}}} & {{equation}\mspace{14mu}\lbrack 3\rbrack}\end{matrix}$

For a given number of coded bits before interleaving, b_(db), each bitmay be denoted by an index, k=0, 1 . . . b_(db)−1. The interleaver block406 may assign bits to the first spatial stream, spatial stream 0,b_(st)(0), for bit indexes k=0, n_(st), 2*n_(st), . . . , b_(db)−n_(st).The interleaver block 406 may assign bits to spatial stream 1,b_(st)(1), for bit indexes k=1, n_(st)+1, 2*n_(st)+1, . . . ,b_(db)−n_(st)+1. The interleaver block 406 may assign bits to spatialstream 2, b_(st)(2), for bit indexes k=2, n_(st)+2, 2*n_(st)+2, . . . ,b_(db)−n_(st)+2. The interleaver block 406 may assign bits to spatialstream n_(st), b_(st)(n_(st)), for bit indexes k=n_(st)1, 2*n_(st)−1,3*n_(st)−1, . . . , b_(db)−1.

The plurality of mapper blocks 408 a . . . 408 n may comprise a numberof individual mapper blocks that is equal to the number of individualstreams generated by the interleaver block 406. Each individual mapperblock 408 a . . . 408 n may receive a plurality of bits from acorresponding individual stream, mapping those bits into a “symbol” byapplying a modulation technique based on a “constellation” utilized totransform the plurality of bits into a signal level representing thesymbol. The representation of the symbol may be a complex quantitycomprising in-phase (I) and quadrature (Q) components. The mapper block408 a . . . 408 n for stream i may utilize a modulation technique to mapa plurality of bits, b_(st)(i), into a symbol.

The beamforming V matrix block 412 may apply the beamforming techniqueto the plurality of symbols, or “spatial modes”, generated from theplurality of mapper blocks 408 a . . . 408 n. The beamforming V matrixblock 412 may generate a plurality of signals where the number ofsignals generated may be equal to the number of transmitting antenna atthe transmitter 400. Each signal in the plurality of signals generatedby the beamforming V block 412 may comprise a weighted sum of at leastone of the received symbols from the mapper blocks 408 a . . . 408 n.

The plurality of IFFT blocks 410 a . . . 410 n may receive a pluralityof signals from the beamforming block 412. Each IFFT block 410 a . . .410 n may subdivide the bandwidth of the RF channel into a plurality ofn sub-band frequencies to implement orthogonal frequency divisionmultiplexing (OFDM), buffering a plurality of received signals equal tothe number of sub-bands. Each buffered signal may be modulated by acarrier signal whose frequency is based on that of one of the sub-bands.Each of the IFFT blocks 410 a . . . 410 n may then independently sumtheir respective buffered and modulated signals across the frequencysub-bands to perform an n-point IFFT thereby generating a composite OFDMsignal.

The plurality of digital (D) to analog (A) conversion and antenna frontend blocks 414 a . . . 414 n may receive the plurality of signalsgenerated by the plurality of IFFT blocks 410 a . . . 410 n. The digitalsignal representation received from each of the plurality of IFFT blocks410 a . . . 410 n may be converted to an analog RF signal that may beamplified and transmitted via an antenna. The plurality of D to Aconversion and antenna front end blocks 414 a . . . 414 n may be equalto the number of transmitting antenna 415 a . . . 415 n. Each D to Aconversion and antenna front end block 414 a . . . 414 n may receive oneof the plurality of signals from the beamforming V matrix block 412 andmay utilize an antenna 415 a . . . 415 n to transmit one RF signal viaan RF channel.

In the receiver 401, the plurality of antenna front end and A to Dconversion blocks 416 a . . . 416 n may receive analog RF signals via anantenna, converting the RF signal to baseband and generating a digitalequivalent of the received analog baseband signal. The digitalrepresentation may be a complex quantity comprising I and Q components.The number of antenna front end and A to D conversion blocks 416 a . . .416 n may be equal to the number of receiving antenna 417 a . . . 417 n.

The plurality of FFT blocks 420 a . . . 420 n may receive a plurality ofsignals from the plurality of antenna front end and A to D conversionblocks 416 a . . . 416 n. The plurality of FFT blocks 420 a . . . 420 nmay be equal to the number of antenna front end and A to D conversionblocks 416 a . . . 416 n. Each FFT block 420 a . . . 420 n may receive asignal from an antenna front end and A to D conversion block 416 a . . .416 n, independently applying an n-point FFT technique, and demodulatingthe signal by a utilizing a plurality of carrier signals based on the nsub-band frequencies utilized in the transmitter 400. The demodulatedsignals may be mathematically integrated over one sub band frequencyperiod by each of the plurality of FFT blocks 420 a . . . 420 n toextract n symbols contained in each of the plurality of OFDM signalsreceived by the receiver 401.

The beamforming U* block 418 may apply the beamforming technique to theplurality of signals received from the plurality of FFT blocks 420 a . .. 420 n. The beamforming U* block 418 may generate a plurality ofsignals where the number of signals generated may be equal to the numberof spatial streams utilized in generating the signals at the transmitter400. Each of the plurality of signals generated by the beamforming U*block 418 may comprise a weighted sum of at least one of the signalsreceived from the FFT blocks 420 a . . . 420 n.

The channel estimates block 422 may utilize preamble information,contained in a received RF signal, to compute channel estimates. Theequalizer block 424 may receive signals generated by the beamforming U*block 418. The equalizer block 424 may process the received signalsbased on input from the channel estimates block 422 to recover thesymbol originally generated by the transmitter 400. The equalizer block424 may comprise suitable logic, circuitry, and/or code that may enabletransformation of symbols received from the beamforming U* block 418 tocompensate for fading in the RF channel.

The plurality of demapper blocks 426 a . . . 426 n may receive symbolsfrom the equalizer block 424, reverse mapping each symbol to one or morebinary bits by applying a demodulation technique, based on themodulation technique utilized in generating the symbol at thetransmitter 400. The plurality of demapper blocks 426 a . . . 426 n maybe equal to the number of streams in the transmitter 400.

The deinterleaver block 428 may receive a plurality of bits from each ofthe demapper blocks 426 a . . . 426 n, rearranging the order of bitsamong the received plurality of bits. The deinterleaver block 428 mayrearrange the order of bits from the plurality of demapper blocks 426 a. . . 426 n in, for example, the reverse order of that utilized by theinterleaver 406 in the transmitter 400. The depuncture block 430 mayinsert “null” bits into the output data block received from thedeinterleaver block 428 that were removed by the puncture block 404. TheViterbi decoder block 432 may decode a depunctured output data block,applying a decoding technique that may recover the binary data blocksthat were input to the coding block 402.

In operation, the processor 440 may receive decoded data from theViterbi decoder 432. The processor 440 may communicate received data tothe baseband processor 442 for analysis and further processing. Theprocessor 440 may also communicate data received via the RF channel, bythe receiver 401, to the channel estimates block 422. This informationmay be utilized by the channel estimates block 422, in the receiver 401,to compute channel estimates for a received RF channel. The basebandprocessor 442 may generate data to be transmitted via an RF channel bythe transmitter 400. The baseband processor 442 may communicate the datato the processor 440. The processor 440 may generate a plurality of bitsthat are communicated to the coding block 402.

The elements shown in FIG. 4 may comprise components that may be presentin an exemplary embodiment of a wireless communications terminal. Oneexemplary embodiment of a may be a wireless communications transmittercomprising a transmitter 400, a processor 440, and a baseband processor442. Another exemplary embodiment of a may be a wireless communicationsreceiver comprising a receiver 401, a processor 440, and a basebandprocessor 442. Another exemplary embodiment of a may be a wirelesscommunications transceiver comprising a transmitter 400, a receiver 401,a processor 440, and a baseband processor 442.

A MIMO system may transmit information via one or more RF channels asspecified by IEEE standard 802.11n. Such a MIMO system may be referredto as an IEEE 802.11n WLAN device. IEEE 802.11n provides specificationfor 20 MHz channels and 40 RF MHz channels. A 20 MHz RF channel may becharacterized by a bandwidth of about 20 MHz. A 40 MHz RF channel may becharacterized by a bandwidth of about 40 MHz. A MIMO system thattransmits and/or receives information via a 20 MHz RF channel mayutilize a total of 56 subcarriers in the 20 MHz channel, comprising 2pilot tones, and 54 data-bearing subcarriers. The subcarriers may bedistributed symmetrically around a frequency that represents a centerfrequency for a 20 MHz channel, for example. The frequency spacingbetween subcarriers in an IEEE 802.11n WLAN device may be approximatelyequal to 312.5 KHz, for example. A 20 MHz channel may comprise aplurality of subcarriers for which the frequency of a subcarrier,f_(sc)(i), may be represented as:

f _(sc)(i)=f _(center) +iΔ _(f) where,  equation [4]

the frequency, f_(center), may represent the center frequency in a 20MHz channel, the frequency increment, Δ_(f), may represent the frequencyspacing between subcarriers, and the value of the subcarrier index, i,may comprise a plurality of integer values represented as:

0<i≦N _(sc)/2, or  equation [5]

−N _(sc)/2≦i<0, where  equation [6]

N_(sc) may represent the number of subcarriers present in a 20 MHzchannel.

An IEEE 802.11n 40 MHz channel may comprise a plurality of subcarriersfor which the frequency of a subcarrier f⁴⁰ _(sc)(i) may be representedas:

f _(sc) ⁴⁰(i)=f _(primary) +iΔ _(f), or  equation [7]

f _(sc) ⁴⁰(i)=f _(secondary) +iΔ _(f), where  equation [8]

f_(primary) may represent the center frequency of a primary 20 MHzchannel, f_(secondary) may represent the center frequency of a secondary20 MHz channel, and the index, i, may be as defined in equations [7]and[8]. The primary and secondary 20 MHz channels may be adjacent channelssuch that:

f _(secondary) =f _(primary)±20 MHz, where  equation [9]

the secondary 20 MHz channel may be located at an adjacent channel forwhich the center frequency f_(secondary) is either 20 MHz higher or 20MHz lower than the center frequency of the primary 20 MHz channelf_(primary). A 40 MHz channel may comprise a plurality of N_(sc)subcarriers located at the primary 20 MHz channel, and subsequentplurality of N_(sc) subcarriers located at the secondary 20 MHz channel,where N_(sc) may represent the number of subcarriers in a 20 MHzchannel. In this regard, a 40 MHz channel may comprise a total of2N_(sc) subcarriers, where the value N_(sc) may be defined based on thenumber of subcarriers contained in the primary or secondary 20 MHzchannel. If the primary or secondary channel comprises 56 subcarriers,then the 40 MHz channel may comprise a total of 112 subcarriers.

A MIMO system may utilize an antenna configuration represented asN_(TX)×N_(RX), where N_(TX) may represent the number of transmittingantenna at a station. Transmitting antennas may be utilized to transmitsignals via an RF channel. N_(RX) may represent the number of receivingantenna at a station that receives the signals.

The MIMO system may utilize OFDM to encoded one or more bits ofinformation in one or more subcarriers associated with an RF channel.The encoded bits in one subcarrier may be referred to as a symbol. Theencoded bits in the subcarriers associated with an RF channel maycollectively be referred to as an OFDM symbol. The number of binary bitscontained in an OFDM symbol, N_(DBPS), may be determined as shown in thefollowing equation:

N _(DBPS) =N _(DSC) *N _(B)(CON)*R  equation [10]

where N_(DSC), may represent the number of data bearing subcarriers inan RF channel, N_(B)(CON) may represent the number of binary bits/symbolbased on the modulation type, and R may represent the coding rate. For a20 MHz RF channel, N_(DSC) may be approximately equal to 54. For a 40MHz RF channel, N_(DSC) may be approximately equal to 108. For amodulation type of 64 QAM, the number of binary bits/symbol mayapproximately equal to 6. For example, for a 40 MHz RF channel utilizing64 QAM and a ¾ coding rate, N_(DSC)=108, N_(B)(CON)=6, and R=¾.Consequently, the number of bits contained in an OFDM symbolN_(DBPS)=486 bits. Given a transmission rate of about 250,000 OFDMsymbols/second, the corresponding data rate associated with the RFchannel may be more than 121 Mbits/second. When beamforming is utilized,the total number of bits simultaneously transmitted via N_(SS) number ofspatial streams may equal approximately N_(SS)×N_(DBPS). Therefore, fora MIMO system that utilizes 4 transmitting antennas, each of whichtransmits an independent spatial stream, the total number of bitstransmitted approximately simultaneously may equal about 1,944. Thecorresponding data rate associated with the MIMO system may be about 486M bits/second.

Various embodiments of the invention comprise a method for reducing thequantity of information, as measured in a number of bits for example,that may be transmitted to describe a downlink RF channel in explicitfeedback information when compared to an amount of information that maybe transmitted utilizing conventional approaches. The number of bitsthat may be transmitted in explicit feedback information when utilizinga polar coordinate format may be fewer than a corresponding number ofbits that may be transmitted when utilizing a Cartesian coordinateformat.

The feedback V matrix may comprise an array of magnitude andcorresponding angle component pairs associated with beamforming factors.For example, a 3×3 V matrix comprising 3 rows an 3 columns may berepresented as follows:

$\begin{matrix}\begin{bmatrix}{V_{1,1}\left( f_{k} \right)} & {V_{1,2}\left( f_{k} \right)} & {V_{1,3}\left( f_{k} \right)} \\{V_{2,1}\left( f_{k} \right)} & {V_{2,2}\left( f_{k} \right)} & {V_{2,3}\left( f_{k} \right)} \\{V_{3,1}\left( f_{k} \right)} & {V_{3,2}\left( f_{k} \right)} & {V_{3,3}\left( f_{k} \right)}\end{bmatrix} & {{equation}\mspace{14mu}\lbrack 11\rbrack}\end{matrix}$

where V_(m,n)(f_(k)) may represent a beamforming factor associated withthe k^(th) frequency subcarrier transmitted by the m^(th) transmittingantenna and received by an n^(th) receiving antenna, and k=1,2, . . . ,N_(sc). The value N_(sc) may represent the number of frequencysubcarriers including all data tones and pilot tones associated with anRF channel.

In one aspect related to the matrix V, a matrix multiplication of V andV* may produce an identity I matrix. The matrix V* may represent anHermitian transpose of the matrix V. An Hermitian transpose of a matrixmay be represented by constructing a transpose of the matrix followed byconstructing a complex conjugate version of each element in thetransposed matrix. An identity matrix may be characterized by an arrayfor which diagonal elements contain a value equal to 1 and non-diagonalelements contain a value equal to 0. An identity matrix may berepresented as a square matrix in that the number of rows in the matrixmay be equal to the number of columns. When the conditions VV*=I andV*V=I are true, the matrix V may be referred to as a unitary matrix. Theunitary matrix V may contain redundant information. Consequently, thenumber of degrees of freedom of the matrix V may be less than N×N. Thismay imply that it may be possible to reduce the number of parametersbased on the number of degrees of freedom of the matrix V, for example,using angles.

When the V matrix contains redundant information, at least a portion ofthe angle components contained in the matrix may be redundant. In thiscase, the number of bits that may be transmitted as explicit feedbackinformation in a redundancy reduced V matrix may be reduced whencompared to conventional approaches that may not utilize redundantinformation.

Various embodiments of the invention may comprise a method and systemfor reducing the number of bits that may be transmitted in explicitfeedback information. The method and system may utilize Givens rotation.

An N×M matrix comprising matrix elements arranged in N rows and Mcolumns may represent a N-dimensional vector space. For a given matrixelement for which a corresponding value is not equal to 0, a Givensrotation may be utilized to rotate the N-dimensional vector space suchthat the value of the corresponding matrix element in the rotated vectorspace is about equal to 0. The Givens rotation may be represented as anN×N Givens rotation matrix, G_(li)(ψ), in the following expression:

$\begin{matrix}{{G_{li}\left( \psi_{li} \right)} = \begin{bmatrix}I_{1 - 1} & 0 & 0 & 0 & 0 \\0 & {\cos \left( \psi_{li} \right)} & 0 & {\sin \left( \psi_{li} \right)} & 0 \\0 & 0 & I_{i - 1 - 1} & 0 & 0 \\0 & {- {\sin \left( \psi_{li} \right)}} & 0 & {\cos \left( \psi_{li} \right)} & 0 \\0 & 0 & 0 & 0 & I_{N - i}\end{bmatrix}} & {{equation}\mspace{14mu}\lbrack 12\rbrack}\end{matrix}$

where I may indicate a row in a matrix, i may indicate a column, I_(n)may represent an identity matrix comprising n rows and n columns, andψ_(li) may represent a rotation angle. The value of the element locatedat the I^(th) row and I^(th) column in the matrix of equation [12] maybe about equal to the value of the expression cos(ψ_(li)). The value ofthe element located at the i^(th) row and i^(th) column may be aboutequal to the value of the expression cos(ψ_(li)). The value of theelement located at the I^(th) row and i^(th) column may be about equalto the value of the expression −sin(ψ_(li)). The value of the elementlocated at the i^(th) row and I^(th) column may be about equal to thevalue of the expression sin(ψ_(li)). Each “0” indicated in the matrix ofequation [12] may represent one or more matrix elements for which eachcorresponding value is about 0. For example, the first row of 0's in thematrix may represent a block of elements comprising I−1 rows and N-(I−1)columns for which the value of each element is about 0. A block ofelements comprising 0 rows and/or 0 columns may be omitted from thematrix of equation [12].

A Givens rotation matrix may generally be utilized to rotate elements ina matrix for which the associated values are represented by realnumbers. The beamforming matrix, V, may generally comprise elements forwhich the associated values are represented by complex numbers. Thematrix V may be multiplied by an M×M column-wise phase shift matrix,{tilde over (D)}, which may be represented as follows:

$\begin{matrix}{\overset{\sim}{D} = \begin{bmatrix}^{j\; \theta_{1}} & 0 & \ldots & 0 \\0 & ^{j\; \theta_{2}} & 0 & \vdots \\\vdots & 0 & \ddots & 0 \\0 & \ldots & 0 & ^{j\; \theta_{M}}\end{bmatrix}} & {{equation}\mspace{14mu}\lbrack 13\rbrack}\end{matrix}$

where θ_(i) may represent a phase shift associated with column i, e mayrepresent a value about equal to 2.718, and j may represent the complexvalue equal to the square root of −1.

The matrix {tilde over (V)}=V{tilde over (D)} may represent acolumn-wise phase shifted version of the matrix V. The correspondingvalues of elements located at the i^(th) row and i^(th) column of thematrix {tilde over (V)}, multiplied by series of G_(li) and D_(i), maybe represented by real numbers. The unitary matrix V may be column-wisephase invariant. For a column-wise phase invariant matrix, the matrix Vmay be considered to be equivalent to the matrix {tilde over (V)}.

The matrix V may be multiplied by an N×N second phase shift matrix,D_(i), which may be represented as follows:

$\begin{matrix}{D_{i} = \begin{bmatrix}I_{i} & 0 & \ldots & 0 \\0 & ^{j\; \phi_{{i + 1},i}} & 0 & \vdots \\\vdots & 0 & \ddots & 0 \\0 & \ldots & 0 & ^{j\; \phi_{N,i}}\end{bmatrix}} & {{equation}\mspace{14mu}\lbrack 14\rbrack}\end{matrix}$

where φ_(a,b) may represent a phase shift associated with a matrixelement located at row a and column b in {tilde over (V)}.

The result of the matrix multiplication D_(i)*V{tilde over (D)} mayrepresent a phase shifted version of the matrix V, where D_(i)* mayrepresent an Hermitian transpose of the phase shift matrix D_(i). Thecorresponding values of elements located at the i^(th) column of theproduct matrix D_(i)*V{tilde over (D)} may be represented by realnumbers.

The matrix product G_(li)(ψ_(li))D_(i)*V{tilde over (D)} may represent aGivens rotated version of the matrix D_(i)*V{tilde over (D)}. AnN-dimensional vector space represented by the matrix productG_(li)(ψ_(li))D_(i)*V{tilde over (D)} may be equivalent to a ψ_(li)degree angular rotation applied to the N-dimensional vector spacerepresented by the matrix D_(i)*V{tilde over (D)}. If a nonzero value isassociated with an element located in the I^(th) row and i^(th) columnof the matrix D_(i)*V{tilde over (D)}, represented as the (l,i) element,the corresponding value of the (l,i) element of the matrix productG_(li)(ψ_(li))D_(i)*V{tilde over (D)} may be about equal to 0.

One characteristic of Givens rotations is that rotations may be appliediteratively to select matrix elements for which corresponding values areto be set to about 0 as a result of successive rotations. For example,based on a current Givens rotation matrix, G_(li)(ψ_(li)), and asubsequent Givens matrix G_(ki)(ψ_(ki)), a subsequent matrix productG_(li)(ψ_(li))G_(ki)(ψ_(ki))D_(i)*V{tilde over (D)} may be computed. Thevalue of the (l,i) element of the subsequent matrix product may be equalto about 0. The value of the (k,i) element of the subsequent matrixproduct may also be about equal to 0.

In various embodiments of the invention, the process may be repeateduntil a subsequent rotated matrix is computed such that the valueassociated with each element I in column i is about equal to 0. Therange of values for the row indicator I may be subject to the followingcondition:

N≧1>i  equation [15]

where N may represent the number of rows in the matrix V.

After repeating the process for each element I in column i=1, forexample, a subsequent rotated matrix may be represented as follows:

$\begin{matrix}{{{G_{21}\left( \psi_{21} \right)}{G_{31}\left( \psi_{31} \right)}\mspace{20mu} \ldots \mspace{14mu} {G_{N\; 1}\left( \psi_{N\; 1} \right)}D_{1}^{*}V\; \overset{\sim}{D}} = {\quad\begin{bmatrix}1 & 0 & \; & \ldots & 0 \\0 & \; & \; & \; & \; \\\vdots & \; & V_{2} & \; & \; \\0 & \; & \; & \; & \;\end{bmatrix}}} & {{equation}\mspace{14mu}\lbrack 16\rbrack}\end{matrix}$

where V₂ may represent an (N−1)×(M−1) submatrix.

For an i^(th) column, equation [16] may be generalized and representedas follows:

$\begin{matrix}{{{G_{{i + 1},i}\left( \psi_{21} \right)}{G_{{i + 2},i}\left( \psi_{31} \right)}\mspace{14mu} \ldots \mspace{14mu} {G_{N,1}\left( \psi_{n\; 1} \right)}D_{1}^{*}R_{i}} = {\quad\begin{bmatrix}I_{i} & 0 & \; & \ldots & 0 \\0 & \; & \; & \mspace{11mu} & \; \\\vdots & \mspace{11mu} & V_{i + 1} & \; & \; \\0 & \; & \; & \; & \;\end{bmatrix}}} & {{equation}\mspace{14mu}\lbrack 17\rbrack}\end{matrix}$

where R_(i) may represent a previous rotated matrix, and V₁₊₁ mayrepresent an (N−i)×(M−i) submatrix. The phase shift angle θ_(i+1) inequation [13] may be defined such that the (1,1) element in the matrixV_(i+1) may be represented by real numbers.

In various embodiments of the invention, the process shown in equation[17] may be repeated for a subsequent column wherein the range of valuesfor the column indicator i may be subject to the following condition:

min(M, N−1)≧i≧1  equation [18]

where N may represent the number of rows in the matrix V, M mayrepresent the number of columns in the matrix V, and min(a,b) mayrepresent an expression, the value of which may be the minimum valueamong arguments a and b.

Based on equations [12], [14], [17], [18], and the unitary matrixproperty, the matrix V may be expressed based the Givens rotationsaccording to the following equation:

$\begin{matrix}{V = {\sum\limits_{i = 1}^{m\; i\; {n{({M,{N - 1}})}}}{\left\lbrack {D_{i}{\prod\limits_{l = {i + 1}}^{N}{G_{li}^{*}\left( \psi_{li} \right)}}} \right\rbrack \times \overset{\sim}{I}}}} & {{equation}\mspace{14mu}\lbrack 19\rbrack}\end{matrix}$

where Ī may represent an N×M matrix comprising an identity matrixI_(min(N,M)) and either a block of max(0,N−M) rows of elements, or ablock of max(0,M−N) columns of elements, with each element of a valueabout 0, max(a,b) may represent an expression, the value of which may bethe maximum value among arguments a and b, and the operator π[X_(i)] mayrepresent a right-side matrix multiplication product among a set ofmatrices X_(i). G_(li)(ψ_(li))* may represent an Hermitian transpose ofthe Givens rotation matrix G_(li)(o_(i)). The Givens rotation matrixG_(li)(ψ_(li)) is as defined in equation [12] and the phase shift matrixD_(i) is as defined in equation [14].

In various embodiments of the invention, explicit feedback informationmay comprise values for one or more rotation angle parameters comprisingone or more phase shift angles φ_(li) and one or more Givens rotationangles ψ_(li). The following table represents the number of parametersthat may be utilized in explicit feedback information to represent anN×M beamforming matrix V, for exemplary values of N rows and M columns:

TABLE 1 Angles to be reported in explicit feedback informationDimensions Number of (N × M) Parameters Explicit Feedback AngleParameters 2 × 1 2 Φ₂₁, ψ₂₁ 2 × 2 2 Φ₂₁, ψ₂₁ 3 × 1 4 Φ₂₁, Φ₃₁, ψ₂₁, ψ₃₁3 × 2 6 Φ₂₁, Φ₃₁, ψ₂₁, ψ₃₁, Φ₃₂, ψ₃₂ 3 × 3 6 Φ₂₁, Φ₃₁, ψ₂₁, ψ₃₁, Φ₃₂,ψ₃₂ 4 × 1 6 Φ₂₁, Φ₃₁, Φ₄₁, ψ₂₁, ψ₃₁, ψ₄₁ 4 × 2 10 Φ₂₁, Φ₃₁, Φ₄₁, ψ₂₁,ψ₃₁, ψ₄₁, Φ₃₂, Φ₄₂, ψ₃₂, ψ₄₂ 4 × 3 12 Φ₂₁, Φ₃₁, Φ₄₁, ψ₂₁, ψ₃₁, ψ₄₁, Φ₃₂,Φ₄₂, ψ₃₂, ψ₄₂, Φ₄₃, ψ₄₃ 4 × 4 12 Φ₂₁, Φ₃₁, Φ₄₁, ψ₂₁, ψ₃₁, ψ₄₁, Φ₃₂, Φ₄₂,ψ₃₂, ψ₄₂, Φ₄₃, ψ₄₃

The feedback angles represented in Table 1 may be reported for each ofthe N_(sc) plurality of frequency subcarriers associated with thecorresponding downlink RF channel. The order of angles represented inTable 1 may be determined based on the order of multiplications that maybe performed when reconstructing the V matrix for each subcarrier at theWLAN 106 that receives the explicit feedback information. For example,for given feedback angles, ψ_(li) and ψ_(li), the angles may berepresented as a vectors of angles, φ_(li)[−N_(sc)/2, −N_(sc)/2+1, . . ., −2, −1, 1, 2, . . . , N_(sc)/2−1, N_(sc)/2] and ψ_(li)[−N_(sc)/2,−N_(sc)/2+1, . . . , −2, −1, 1, 2, . . . , N_(sc)/2−1, N_(sc)/2], whereN_(sc) may represent the number of subcarriers associated with thedownlink RF channel.

Actual values for angles reported in explicit feedback information maybe determined based on a codebook. For example, a range of true valuesfor Givens rotation angles may comprise 0≦ψ_(li)≦π/2. A range of truevalues for phase rotation angles may comprise 0≦φ_(li)≦2π, for example.Values for the Givens rotation angles may be approximated based on abinary encoding comprising a plurality of b_(ψ) bits. Values for Givensrotation angles that are represented by a binary encoding of bits may bereferred to as quantized values. The range of values for binaryquantized values for the Givens rotation angles may comprise kπ/2^(b)^(ψ) ⁺¹ +π/s ^(b) ^(ψ) ⁺², where k may be equal to one of a range ofvalues comprising 0,1, . . . , 2^(b) ^(ψ) −1, for example. Values forthe phase rotation angles may be approximated based on a binary encodingcomprising a plurality of b_(φ) bits. The range of values for binaryquantized values for the phase rotation angles may comprise kπ/2^(b)^(ψ) ⁻¹+π/2^(b) ^(ψ) , where k may be equal to one of a range of valuescomprising 0,1, . . . , 2^(b) ^(ψ) −1, for example. Differences in valuebetween a true value and a corresponding quantized value may be referredto as a quantization error. The potential size of a given quantizationerror may be based on the number of bits utilized during binaryencoding.

The number of bits utilized during binary encoding of ψ_(li) and duringbinary encoding of ψ_(li) respectively may be represented as a (b_(ψ),b_(φ)) tuple. A tuple utilized in an exemplary codebook may comprise(1,3) to represent the combination b₁₀₄ =1 and b_(φ)=3. Other tuples inthe exemplary codebook may comprise (2,4), (3,5), or (4,6).

Various embodiments of the invention may comprise a method and a systemfor utilizing Givens rotations for representing beamforming matrices inexplicit feedback information. In one aspect of the invention, thenumber of bits contained in the explicit feedback information may bereduced in comparison to a corresponding number of bits contained inexplicit feedback information based on a Cartesian coordinate formatrepresentation. The number of bits contained in explicit feedbackinformation, which is generated in accordance with an embodiment of theinvention may be determined based on the size of the V matrix, asmeasured by N rows and M columns, the number of subcarriers associatedwith the corresponding downlink RF channel, and the number of bitsutilized during binary encoding of the Givens rotation angles, and phaserotation angles.

Table 2 presents an exemplary mapping between bits contained in explicitfeedback information, and the corresponding angle that may be reported.In Table 2, N=2, M=2, the downlink channel may be a 20 MHz RF channel,and the bits utilized may be represented by (3,5). The range b_(j) . . .b_(k) may represent a range of bit positions in the explicit feedbackinformation where in b_(j) may represent the least significant bit (LSB)in the range and b_(k) may represent the most significant bit (MSB) inthe range, for example. The notation f_(p) may represent a frequencysubcarrier associated with the RF channel. Values for the index p mayrepresent a range of integer values as determined in equations [5] and[6]. The notation ψ_(li)(f_(p)) may refer to a Givens rotation angleassociated with frequency subcarrier f_(p). The notation ψ_(li)(f_(p))may refer to a phase rotation angle associated with frequency subcarrierf_(p).

TABLE 2 Exemplary encoding of explicit feedback information FeedbackBits Angle Reported b₁ . . . b₅ Φ₂₁(f⁻²⁸) b₆ . . . b₁₀ Φ₂₁(f⁻²⁷) . . . .. . b₂₇₆ . . . b₂₈₀ Φ₂₁(f₂₈) b₂₈₁ . . . b₂₈₃ ψ₂₁(f⁻²⁸) b₂₈₄ . . . b₂₈₆ψ₂₁(f⁻²⁷) . . . . . . b₄₄₆ . . . b₄₄₈ ψ₂₁(f₂₈)

The exemplary explicit feedback information shown in Table 2 contains448 bits: 168 bits may be utilized to communicate Givens rotationangles, and 280 bits may be utilized to communicate phase rotationangles.

Table 3 presents exemplary comparisons between the number of bytescontained in explicit feedback information, which is generated inaccordance with an embodiment of the invention, and a correspondingnumber of bytes contained in explicit feedback information such that thePER performance may be within about 0.5dB of the full resolution, ortrue values for Givens and/or phase rotation angles, PER performance forbeamforming, which is generated based on a Cartesian coordinate formatrepresentation. The comparison in Table 3 may be based on a 20 MHz typeE RF channel as specified in IEEE 802.11. The first row in Table 3represents a number of bytes contained in explicit feedback information,which is generated based on a Cartesian coordinate formatrepresentation. The second row in Table 3 represents a number of bytescontained in explicit feedback information such that the PER performancemay be within about 0.5dB of the full resolution, or true values forGivens and/or phase rotation angles, PER performance for beamforming,which is generated based on Givens rotations in accordance with anembodiment of the invention. The first column in Table 3 represents asize of a V matrix where N=2 and M=2. The second column in Table 3represents a size of a V matrix where N=3 and M=3. The third column inTable 3 represents a size of a V matrix where N=4 and M=2. The fourthcolumn in Table 3 represents a size of a V matrix where N=4 and M=4.

When N=2 and M=2, the Cartesian coordinate format representation,indicated in the first row of Table 3, may be utilized to report 4 setsof I and Q amplitude component values in explicit feedback informationwhere each set comprises I and Q amplitude component values associatedwith each frequency subcarrier associated with the 20 MHz RF channel.The I and Q amplitudes may be quantized and represented in a 10 bitbinary word, for example utilizing 4 bits to quantize each I amplitudeand 6 bits to quantize each Q amplitude. The Cartesian coordinate formatrepresentation of the explicit feedback information may comprise 280bytes. The Givens rotation version, indicated in the second row of Table3, may be utilized to report one set of Givens rotation angles and phaserotation angles, as indicated in Table 1. The Givens rotation versionmay utilize 3 bits to quantize each Givens rotation angle and 5 bits toquantize each phase rotation angle, for example. The Givens rotationversion of the explicit feedback information may comprise 56 bytes.

When N=3 and M=3, the Cartesian coordinate format representation may beutilized to report 9 sets of I and Q amplitude component values inexplicit feedback information. The I and Q amplitudes may be quantizedand represented in a 8 bit binary word, for example utilizing 3 bits toquantize each I amplitude and 5 bits to quantize each Q amplitude. TheCartesian coordinate format representation of the explicit feedbackinformation may comprise 504 bytes. The Givens rotation version,indicated in the second row of Table 3, may be utilized to report 3 setsof Givens rotation angles and phase rotation angles as indicated inTable 1. The Givens rotation version may utilize 3 bits to quantize eachGivens rotation angle and 5 bits to quantize each phase rotation angle,for example. The Givens rotation version of the explicit feedbackinformation may comprise 168 bytes.

When N=4 and M=2, the Cartesian coordinate format representation may beutilized to report 8 sets of I and Q amplitude component values inexplicit feedback information. The I and Q amplitudes may be quantizedand represented in a 8 bit binary word, for example utilizing 3 bits toquantize each I amplitude and 5 bits to quantize each Q amplitude. TheCartesian coordinate format representation of the explicit feedbackinformation may comprise 448 bytes. The Givens rotation version,indicated in the second row of Table 3, may be utilized to report 5 setsof Givens rotation angles and phase rotation angles as indicated inTable 1. The Givens rotation version may utilize 2 bits to quantize eachGivens rotation angle and 4 bits to quantize each phase rotation angle,for example. The Givens rotation version of the explicit feedbackinformation may comprise 210 bytes.

When N=4 and M=4, the Cartesian coordinate format representation may beutilized to report 16 sets of I and Q amplitude component values inexplicit feedback information. The I and Q amplitudes may be quantizedand represented in a 8 bit binary word, for example utilizing 3 bits toquantize each I amplitude and 5 bits to quantize each Q amplitude. TheCartesian coordinate format representation of the explicit feedbackinformation may comprise 896 bytes. The Givens rotation version,indicated in the second row of Table 3, may be utilized to report 6 setsof Givens rotation angles and phase rotation angles as indicated inTable 1. The Givens rotation version may utilize 3 bits to quantize eachGivens rotation angle and 5 bits to quantize each phase rotation angle,for example. The Givens rotation version of the explicit feedbackinformation may comprise 336 bytes. Alternatively, the Givens rotationversion may utilize 2 bits to quantize each Givens rotation angle and 4bits to quantize each phase rotation angle, for example. In this case,the Givens rotation version of the explicit feedback information maycomprise 252 bytes.

TABLE 3 Quantity of Explicit Feedback Information for 20 MHz Type EChannel (Bytes) 2 × 2 3 × 3 4 × 2 4 × 4 Cartesian 280 504 448 896 Givens56 168 210 336 (252)

Table 4 presents exemplary comparisons between the number of bytescontained in explicit feedback information, which is generated inaccordance with an embodiment of the invention, and a correspondingnumber of bytes contained in explicit feedback information such that thePER performance may be within about 0.5dB of the full resolution, ortrue values for Givens and/or phase rotation angles, PER performance forbeamforming, which is generated based on a Cartesian coordinate formatrepresentation. Table 4 represents equivalent information from Table 3where the RF channel is 40 MHz. The comparison in Table 4 may be basedon a 40 MHz type E RF channel as specified in IEEE 802.11. The first rowin Table 4 represents a number of bytes contained in explicit feedbackinformation, which is generated based on a Cartesian coordinate formatrepresentation. The second row in Table 4 represents a number of bytescontained in explicit feedback information such that the PER performancemay be within about 0.5dB of the full resolution, or true values forGivens and/or phase rotation angles, PER performance for beamforming,which is generated based on Givens rotations in accordance with anembodiment of the invention. The first column in Table 4 represents asize of a V matrix where N=2 and M=2. The second column in Table 4represents a size of a V matrix where N=3 and M=3. The third column inTable 4 represents a size of a V matrix where N=4 and M=2. The fourthcolumn in Table 4 represents a size of a V matrix where N=4 and M=4.

TABLE 4 Quantity of Explicit Feedback Information for 40 MHz Type EChannel (Bytes) 2 × 2 3 × 3 4 × 2 4 × 4 Cartesian 570 1,026 912 1,824Givens 114 342 428   684 (513)

FIG. 5 is a flowchart illustrating exemplary steps for utilizing Givensrotations for beamforming matrices in explicit feedback information, inaccordance with an embodiment of the invention. Referring to FIG. 5, isstep 502 a column counter, i, may be initialized to a value of 1. Instep 504, a signal Y, which was transmitted by a WLAN station 106, maybe received by a WLAN station 104 via a downlink RF channel. In step 506the receiving WLAN station 104 may compute a beamforming V matrix. The Vmatrix may comprise N rows and M columns.

In step 508, a row counter, I, may be initialize to a value based on thecurrent value of the row counter i. In step 510, a residual matrix{tilde over (V)} may be computed by phase shifting the beamforming Vmatrix such that the (1,1) element in the V_(i) matrix may berepresented a real number. In step 512, a current residual matrix may bephase shifted such that elements in the i^(th) column may be representedby real numbers. In step 516, a Givens rotation may be applied such thatthe value of the element (l,i) is about 0 in the Givens rotated matrix.Step 518 may determine if the current value of the row counter is equalto the value N. If the current value of the row counter is not equal tothe value N, in step 520, the value of the row counter may beincremented. Step 516 may follow step 520. If the current value of therow counter is equal to the value N, step 522 may determine if thecurrent value of the column counter is about equal to the value of theexpression min(M,N−1). If the current value of the column counter is notapproximately equal to the value of the expression min(M,N−1), step 524may increment the value of the column counter. Step 508 may follow step524. If the current value of the column counter is about equal to thevalue of the expression min(M,N−1), in step 526, values for Givensrotation angles and phase rotation angles may be encoded. The values ofthe angles may be quantized and encoded based on a codebook, forexample. In step 528, explicit feedback information may be transmittedfrom the WLAN station 104 to the WLAN station 106 via an uplink RFchannel.

Aspects of a system for processing signals in a communication system maycomprise a processor 282 that enables computation of a beamformingmatrix for a signal received from a transmitter 302 in a MIMOcommunication system. The processor 282 may enable rotation, at areceiver 322 in the MIMO communication system, of components of aresidual matrix generated from the computed beamforming matrix utilizinga Givens rotation. The processor 282 may enable a feeding back ofinformation resulting from the rotation to the transmitter 302 in theMIMO communication system.

The processor 282 may enable a phase shifting of the computedbeamforming matrix to generate the residual matrix. The phase shiftingmay be done such that elements in an i^(th) column of the residualmatrix may be represented by at least one real number. The processor 282may enable a rotation of the phase shifted residual matrix to generate asubsequent residual matrix based on the Givens rotation such that avalue of an element in an l^(th)row in the i^(th) column is about equalto 0.

The Givens rotation may utilize a Givens (l, i) matrix as described inequation [12], for example. A value of an element in an I^(th) row andin the I^(th) column may be about equal to cos(ψ_(li)). A value of anelement in the I^(th) row and in the i^(th) column may be about equal to−sin(ψ_(li)). A value of an element in the i^(th) row and in the I^(th)column may be about equal to sin(ψ_(li)). A value of an element in thei^(th) row and in the i^(th) column may be about equal to cos(ψ_(li)). Avalue for a corresponding rotation angle ψ_(li) may be based on I and i.The value for I may be greater than the value for i.

The processor 282 may enable a rotation of the subsequent residualmatrix based on one or more subsequent Givens rotations and/or one ormore subsequent phase shifting. The subsequent shifting may be done suchthat elements in an m^(th) column of the subsequent residual matrix arerepresented by at least on real number. The processor 282 may enable arotation of the subsequent phase shifted subsequent residual matrixbased on the subsequent Givens rotation such that a value of an elementin an n^(th) row and in the m^(th) column is about equal to0.

The subsequent Givens rotation may utilize a Givens (n, m) matrix. Avalue of an element in an n^(th) row and in the n^(th) column may beabout equal to cos(ψ_(nm)). A value of an element in the n^(th) row andin the m^(th) column may be about equal to −sin(ψ_(nm)). A value of anelement in the m^(th) row and in the n^(th) column may be about equal tosin(ψ_(nm)). A value of an element in the m^(th) row and in the m^(th)column may be about equal to cos(ψ_(nm)). A value for a correspondingrotation angle ψ_(nm) may be based on n and m. The value for n may begreater than the value for m.

Accordingly, the present invention may be realized in hardware,software, or a combination of hardware and software. The presentinvention may be realized in a centralized fashion in at least onecomputer system, or in a distributed fashion where different elementsare spread across several interconnected computer systems. Any kind ofcomputer system or other apparatus adapted for carrying out the methodsdescribed herein is suited. A typical combination of hardware andsoftware may be a general-purpose computer system with a computerprogram that, when being loaded and executed, controls the computersystem such that it carries out the methods described herein.

The present invention may also be embedded in a computer programproduct, which comprises all the features enabling the implementation ofthe methods described herein, and which when loaded in a computer systemis able to carry out these methods. Computer program in the presentcontext means any expression, in any language, code or notation, of aset of instructions intended to cause a system having an informationprocessing capability to perform a particular function either directlyor after either or both of the following: a) conversion to anotherlanguage, code or notation; b) reproduction in a different materialform.

While the present invention has been described with reference to certainembodiments, it will be understood by those skilled in the art thatvarious changes may be made and equivalents may be substituted withoutdeparting from the scope of the present invention. In addition, manymodifications may be made to adapt a particular situation or material tothe teachings of the present invention without departing from its scope.Therefore, it is intended that the present invention not be limited tothe particular embodiment disclosed, but that the present invention willinclude all embodiments falling within the scope of the appended claims.

1. A method for processing signals in a MIMO communication system, themethod comprising: in a transmitter in the MIMO communication system:receiving feedback information from a receiver in said MIMOcommunication system via an uplink RF channel, the feedback informationincluding feedback angles representing a beamforming matrix computed bythe receiver for a downlink RF channel, the feedback angles including atleast one phase shift angle and at least one Givens rotation angle;reconstructing the beamforming matrix using the feedback angles; andutilizing the beamforming matrix for communications with the receiver onthe downlink RF channel.
 2. The method of claim 1, wherein said at leastone phase shift angle represents a phase i^(th) shift of the beamformingmatrix such that elements in an i^(th) column of a residual matrixproduced from phase shifting the beamforming matrix are represented byat least one real number.
 3. The method of claim 2, wherein said atleast one phase shift angle further represents a phase shift of theresidual matrix such that, based on said at least one Givens rotationangle, a value of an element in an I^(th) row and in said i^(th) columnof a subsequent residual matrix is equal to
 0. 4. The method of claim 3,wherein said Givens rotation angle represents a Givens (l, i) matrix:wherein a value of an element in an l^(th) row and in said l^(th) columnis equal to cos(ψ_(ij)); wherein a value of an element in said l^(th)row and in said i^(th) column is equal to −sin(ψ_(li)); wherein a valueof an element in said l_(th) row and in said l_(th) column is equal tosin(ψ_(li)); wherein a value of an element in said l_(th) row and insaid i_(th) column is equal to cos(ψ_(li)); wherein cos( )represents acosine trigonometric function, sin( ) represents a sine trigonometricfunction, and ψ_(li) represents said Givens rotation angle based on saidone or both of said I and said i.
 5. The method of claim 4, wherein saidfeedback information further includes one or both of an additionalGivens rotation angle and an additional phase shift angle.
 6. The methodof claim 5, wherein said additional phase shift angle represents a phaseshift of the subsequent residual matrix such that elements in an m^(th)column of an additional residual matrix are represented by at least onereal number.
 7. The method of claim 6, wherein said additional Givensrotation angle represents a rotation of said additional residual matrixsuch that a value of an element in an nth row and in said m^(th) columnis equal to
 0. 8. The method of claim 7, wherein said additional Givensrotation utilizes a Givens (n, m) matrix: wherein a value of an elementin an n^(th) row and in said n^(th) column is equal to cos(ψ_(nm));wherein a value of an element in an nth row and in said m^(th) column isequal to −sin(ψ_(nm)); wherein a value of an element in an m^(th) rowand in said n^(th) column is equal to sin(ψ_(nm)); wherein a value of anelement in an m^(th) row and in said m^(th) column is equal tocos(ψ_(nm)) wherein cos( ) represents a cosine trigonometric function,sin( ) represents a sine trigonometric function, and ψ_(nm) represents avalue for said additional Givens rotation angle based on said one orboth of said n and said m.
 9. The method of claim 1, wherein saidreconstructing further includes: using an order of said feedback anglesto determine an order of multiplications performed to reconstruct thebeamforming matrix.
 10. The method of claim 1, wherein said feedbackangles are vectors representing values for each subcarrier on thedownlink RF channel.
 11. A transmitter for use in a MIMO communicationsystem, the transmitter comprising: a processor, wherein: said processoris operable to receive feedback information from a receiver in said MIMOcommunication system via an uplink RF channel, the feedback informationincluding feedback angles representing a beamforming matrix computed bythe receiver for a downlink RF channel, the feedback angles including atleast one phase shift angle and at least one Givens rotation angle; saidprocessor is operable to reconstruct the beamforming matrix using thefeedback angles; and said processor is operable to enable thebeamforming matrix to be utilized for communications with the receiveron the downlink RF channel.
 12. The transmitter of claim 11, whereinsaid at least one phase shift angle represents a phase shift of thebeamforming matrix such that elements in an i^(th) column of a residualmatrix produced from phase shifting the beamforming matrix arerepresented by at least one real number.
 13. The transmitter of claim12, wherein said at least one phase shift angle further represents aphase shift of the residual matrix such that, based on said at least oneGivens rotation angle, a value of an element in an I^(th) row and insaid i^(th) column of a subsequent residual matrix is equal to
 0. 14.The transmitter of claim 13, wherein said Givens rotation anglerepresents a Givens (l, i) matrix: wherein a value of an element in anl^(th) row and in said l^(th) column is equal to cos(ψ_(ij)); wherein avalue of an element in said l^(th) row and in said i^(th) column isequal to −sin(ψ_(li)); wherein a value of an element in said i_(th) rowand in said l_(th) column is equal to sin(ψ_(li)); wherein a value of anelement in said i_(th) row and in said i_(th) column is equal tocos(ψ_(li)); wherein cos( ) represents a cosine trigonometric function,sin( ) represents a sine trigonometric function, and ψ_(li) representssaid Givens rotation angle based on said one or both of said I and saidi.
 15. The transmitter of claim 14, wherein said feedback informationfurther includes one or both of an additional Givens rotation angle andan additional phase shift angle.
 16. The transmitter of claim 15,wherein said additional phase shift angle represents a phase shift ofthe subsequent residual matrix such that elements in an m^(th) column ofan additional residual matrix are represented by at least one realnumber.
 17. The transmitter of claim 16, wherein said additional Givensrotation angle represents a rotation of said additional residual matrixsuch that a value of an element in an nth row and in said m^(th) columnis equal to
 0. 18. The transmitter of claim 17, wherein said additionalGivens rotation utilizes a Givens (n, m) matrix: wherein a value of anelement in an n^(th) row and in said n^(th) column is equal tocos(ψ_(nm)); wherein a value of an element in an nth row and in saidm^(th) column is equal to −sin(ψ_(nm)); wherein a value of an element inan m^(th) row and in said n^(th) column is equal to sin(ψ_(nm)); whereina value of an element in an m^(th) row and in said m^(th) column isequal to cos(ψ_(nm)) wherein cos( ) represents a cosine trigonometricfunction, sin( ) represents a sine trigonometric function, and ψ_(nm)represents a value for said additional Givens rotation angle based onsaid one or both of said n and said m.
 19. The transmitter of claim 11,wherein said processor is further operable to use an order of saidfeedback angles to determine an order of multiplications to be performedto reconstruct the beamforming matrix.
 20. The transmitter of claim 11,wherein said feedback angles are vectors representing values for eachsubcarrier on the downlink RF channel.